Reference articles on history, science, culture and more
Surprise meSurprise meSubscribe
Encyclopedia

Appell sequence

Type of polynomial sequence

Wikipedia contributorsAdapted for Fathomly · View authors
2 min read
Source updated January 22, 2026

In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence \{p_{n}(x)\}_{n=0,1,2,\ldots } satisfying the identity

{\frac {d}{dx}}p_{n}(x)=np_{n-1}(x),

and in which p_{0}(x) is a non-zero constant.

Among the most notable Appell sequences besides the trivial example \{x^{n}\} are the Hermite polynomials, the Bernoulli polynomials, and the Euler polynomials. Every Appell sequence is a Sheffer sequence, but most Sheffer sequences are not Appell sequences. Appell sequences have a probabilistic interpretation as systems of moments.

01Equivalent characterizations of Appell sequences

The following conditions on polynomial sequences can easily be seen to be equivalent:

  • For n=1,2,3,\ldots,
{\frac {d}{dx}}p_{n}(x)=np_{n-1}(x)
and p_{0}(x) is a non-zero constant;
  • For some sequence {\textstyle \{c_{n}\}_{n=0}^{\infty } of scalars with c_{0}\neq 0,
p_{n}(x)=\sum _{k=0}^{n}{\binom {n}{k}}c_{k}x^{n-k};
  • For the same sequence of scalars,
p_{n}(x)=\left(\sum _{k=0}^{\infty }{\frac {c_{k}}{k!}}D^{k}\right)x^{n},
where
D={\frac {d}{dx}};
  • For n=0,1,2,\ldots,
p_{n}(x+y)=\sum _{k=0}^{n}{\binom {n}{k}}p_{k}(x)y^{n-k}.

02Recursion formula

Suppose

p_{n}(x)=\left(\sum _{k=0}^{\infty }{c_{k} \over k!}D^{k}\right)x^{n}=Sx^{n},

where the last equality is taken to define the linear operator S on the space of polynomials in x. Let

T=S^{-1}=\left(\sum _{k=0}^{\infty }{\frac {c_{k}}{k!}}D^{k}\right)^{-1}=\sum _{k=1}^{\infty }{\frac {a_{k}}{k!}}D^{k}

be the inverse operator, the coefficients a_{k} being those of the usual reciprocal of a formal power series, so that

Tp_{n}(x)=x^{n}.\,

In the conventions of the umbral calculus, one often treats this formal power series T as representing the Appell sequence p_{n}. One can define

\log T=\log \left(\sum _{k=0}^{\infty }{\frac {a_{k}}{k!}}D^{k}\right)

by using the usual power series expansion of the \log(x) and the usual definition of composition of formal power series. Then we have

p_{n+1}(x)=(x-(\log T)')p_{n}(x).\,

(This formal differentiation of a power series in the differential operator D is an instance of Pincherle differentiation.)

In the case of Hermite polynomials, this reduces to the conventional recursion formula for that sequence.

03Subgroup of the Sheffer polynomials

The set of all Appell sequences is closed under the operation of umbral composition of polynomial sequences, defined as follows. Suppose \{p_{n}(x)\colon n=0,1,2,\ldots \} and \{q_{n}(x)\colon n=0,1,2,\ldots \} are polynomial sequences, given by

p_{n}(x)=\sum _{k=0}^{n}a_{n,k}x^{k}{\text{ and }}q_{n}(x)=\sum _{k=0}^{n}b_{n,k}x^{k}.

Then the umbral composition p\circ q is the polynomial sequence whose nth term is

(p_{n}\circ q)(x)=\sum _{k=0}^{n}a_{n,k}q_{k}(x)=\sum _{0\leq \ell \leq k\leq n}a_{n,k}b_{k,\ell }x^{\ell }

(the subscript n appears in p_{n}, since this is the nth term of that sequence, but not in q, since this refers to the sequence as a whole rather than one of its terms).

Under this operation, the set of all Sheffer sequences is a non-abelian group, but the set of all Appell sequences is an abelian subgroup. That it is abelian can be seen by considering the fact that every Appell sequence is of the form

p_{n}(x)=\left(\sum _{k=0}^{\infty }{\frac {c_{k}}{k!}}D^{k}\right)x^{n},

and that umbral composition of Appell sequences corresponds to multiplication of these formal power series in the operator D.

04Different convention

Another convention followed by some authors (see Chihara) defines this concept in a different way, conflicting with Appell's original definition, by using the identity

{d \over dx}p_{n}(x)=p_{n-1}(x)

instead.

05Hypergeometric Appell polynomials

The enormous class of Appell polynomials can be obtained in terms of the generalized hypergeometric function.

Let \Delta (k,-n) denote the array of k ratios

-{\frac {n}{k}},-{\frac {n-1}{k}},\ldots ,-{\frac {n-k+1}{k}},\quad n\in {\mathbb {N} }_{0},k\in \mathbb {N} .

Consider the polynomial A_{n,p,q}^{(k)}(a,b;m,x)=x^{n}{}_{k+p}F_{q}\left({a_{1}},{a_{2}},\ldots ,{a_{p}},\Delta (k,-n);{b_{1}},{b_{2}},\ldots ,{b_{q}};{\frac {m}{x^{k}}}\right),\quad n,m\in \mathbb {N} _{0},k\in \mathbb {N}

where {}_{k+p}F_{q} is the generalized hypergeometric function.

Theorem. The polynomial family \{A_{n,p,q}^{(k)}(a,b;m,x)\} is the Appell sequence for any natural parameters a,b,p,q,m,k.

For example, if p=0,q=0, k=m, m=(-1)^{k}h{k^{k}} then the polynomials A_{n,p,q}^{(k)}(m,x) become the Gould-Hopper polynomials g_{n}^{m}(x,h) and if p=0,q=0,m=-2,k=2 they become the Hermite polynomials H_{n}(x).

Watch videos about Appell sequenceExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Appell sequence”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Continue exploring

Related topics

Sheffer sequence

In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence : n = 0, 1, 2, 3, ... ) of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics.

Umbral calculus

The term umbral calculus has two related but distinct meanings. In mathematics, before the 1970s, umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to prove them.

Generalized Appell polynomials

In mathematics, a polynomial sequence { p n } {\displaystyle \{p_{n}(z)\}} has a generalized Appell representation if the generating function for the polynomials takes on a certain form: K ( z , w ) = A ( w ) Ψ ( z g ( w ) ) = ∑ n = 0 ∞ p n ( z ) w n {\displaystyle K(z,w)=A(w)\Psi (zg(w))=\sum _{n=0}^{\infty }p_{n}(z)w^{n}} where the generating function or kernel K ( z , w ) {\displaystyle K(z,w)} is composed of the series A ( w ) = ∑ n = 0 ∞ a n w n {\displaystyle A(w)=\sum _{n=0}^{\infty }a_{n}w^{n}\quad } with a 0 ≠ 0 {\displaystyle a_{0}\neq 0} and Ψ ( t ) = ∑ n = 0 ∞ Ψ n t n {\displaystyle \Psi (t)=\sum _{n=0}^{\infty }\Psi _{n}t^{n}\quad } and all Ψ n ≠ 0 {\displaystyle \Psi _{n}\neq 0} and g ( w ) = ∑ n = 1 ∞ g n w n {\displaystyle g(w)=\sum _{n=1}^{\infty }g_{n}w^{n}\quad } with g 1 ≠ 0. {\displaystyle g_{1}\neq 0.} Given the above, it is not hard to show that p n ( z ) {\displaystyle p_{n}(z)} is a polynomial of degree n {\displaystyle n} .